Geometric Topology Laboratory, Assoc. Prof. Erika Kuno
My research theme is mainly mapping class groups of surfaces (including non-orientable surfaces) and 3-dimensional handlebodies. Even familiar objects like one-hole candies and a two‑hole doughnut can give rise to mapping class groups.
I am interested in exploring them from the viewpoints of geometric group theory. Geometric group theory is also applied to the study of large-scale geometry of networks, configuration spaces in robotics, and investigating spaces that appear in data analysis and machine learning.
One of the most important problems in geometric group theory is classifying finitely generated groups by "quasi-isometries". Two finitely generated groups are quasi-isometric if roughly speaking, their word metrics are the same up to linear functions. An interesting part of the geometric group theory is that the properties of the infinite groups are revealed one by one by measuring with a coarse scale of quasi-isometries, but not isometries.
Currently, groups which are quasi-isometric to mapping class groups have hardly been found. Then what I am wondering is the question "Which groups or spaces are quasi-isometric to the mapping class group?". Based on this big theme, I would like to elucidate properties of mapping class groups and deepen their understanding.